Article

A brief review on frictional pressure drop reduction studies for laminar and turbulent flow in helically coiled tubes

Fsadni, Andrew, Whitty, Justin and Stables, Matthew

Available at http://clok.uclan.ac.uk/19602/

Fsadni, Andrew ORCID: 0000-0003-3047-2714, Whitty, Justin ORCID: 0000-0003-1002-5271 and Stables, Matthew ORCID: 0000-0002-9971-9458 (2016) A brief review on frictional pressure drop reduction studies for laminar and turbulent flow in helically coiled tubes. Applied Thermal Engineering, 109 (Part A). pp. 334-343. ISSN 1359-4311

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Accepted Manuscript

A brief review on frictional pressure drop reduction studies for laminar and

turbulent flow in helically coiled tubes

Andrew Michael Fsadni, Justin P.M. Whitty, Matthew A. Stables

PII: S1359-4311(16)31422-3

DOI: http://dx.doi.org/10.1016/j.applthermaleng.2016.08.068

Reference: ATE 8867

To appear in: Applied Thermal Engineering

Received Date: 10 June 2016

Revised Date: 10 August 2016

Accepted Date: 11 August 2016

Please cite this article as: A.M. Fsadni, J.P.M. Whitty, M.A. Stables, A brief review on frictional pressure drop

reduction studies for laminar and turbulent flow in helically coiled tubes, Applied Thermal Engineering (2016), doi:

http://dx.doi.org/10.1016/j.applthermaleng.2016.08.068

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1

1

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Title: A brief review on frictional pressure drop reduction studies for laminar 3

and turbulent flow in helically coiled tubes 4

5

6

7

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Authors: Andrew Michael Fsadni*, Justin P.M. Whitty, Matthew A. Stables 9

10

*Corresponding author 11

12

Contact details: 13

14

Address: University of Central Lancashire, School of Engineering, Rm. KM124, Preston, 15

UK, PR1 2HE 16

17

Email: afsadni@uclan.ac.uk 18

19

Tel: +44 1772893812 20

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A brief review on frictional pressure drop reduction studies for laminar and turbulent 23

flow in helically coiled tubes 24

25

Abstract 26

27

This review, summarises the pertinent literature on drag reduction (DR) in laminar and 28

turbulent flow in coiled tubes. Due to their compact design, ease of manufacture and superior 29

fluid mixing properties, helically coiled tubes are widely used in numerous industries. 30

However, flow through coiled tubes yields enhanced frictional pressure drops and thus, drag 31

reduction is desirable as it can: decrease the system energy consumption, increase the flow 32

rate and reduce the pipe and pump size. The main findings and correlations for the friction 33

factor are summarised for drag reduction with the: injection of air bubbles and addition of 34

surfactant and polymer additives. The purpose of this study is to provide researchers in 35

academia and industry with a concise and practical summary of the relevant correlations and 36

supporting theory for the calculation of the frictional pressure drop with drag reducing 37

additives in coiled tubes. A significant scope for future research has also been identified in 38

the fields of: air bubble and polymer drag reduction techniques. 39

40

Keywords: Helically coiled tube, drag reduction, frictional pressure drop, surfactants, 41

polymer solutions. 42

43

1. Introduction 44

45

Due to their compact design, ease of manufacture and high efficiency in heat and 46

mass transfer, helically coiled tubes are widely used in a number of industries and processes 47

such as in the food, nuclear, aerospace and power generation industries and in heat recovery, 48

refrigeration, space heating and air-conditioning processes. Due to the formation of a 49

secondary flow, which inherently enhances the mixing of the fluid, helically coiled tube heat 50

exchangers are known to yield improved heat transfer characteristics when compared to 51

straight tube heat exchangers. The secondary flow, which finds its origins in the centrifugal 52

force, is perpendicular to the axial fluid direction and reduces the thickness of the thermal 53

boundary layer. However, for single and multiphase flows, the secondary flow yields a 54

substantial increase in the frictional pressure drop, which often results in diminished system 55

efficiencies (due to enhanced pumping power requirements). For air-water two-phase bubbly 56

flow in helically coiled tubes, Akagawa et al. (1971) reported frictional pressure drops in the 57

range of 58

59

60

3

Figure 1: Schematic representation of helical pipe characteristics. 61

1.1 to 1.5 times greater than those in straight tubes, ceteris paribus, whilst, with the use of 62

nanofluids, such a penalty could nullify the enhanced efficiencies gained with the dispersion 63

of nanoparticles in the base fluid (Aly, 2014). Moreover, due to the secondary flow, the flow 64

characteristics are significantly different to those in straight tubes. Whereas in straight tubes 65

the transition from laminar to turbulent flow occurs at Reynolds numbers in the region of 66

2500, the transition in curved tubes takes place at higher Reynolds numbers. The critical 67

Reynolds number (Eq. (1)) is used to determine the transition of the flow from laminar to 68

turbulent flow (Ito, 1959). 69

70 ������ = 24��.�� (1) 71

72

where δ is the curvature ratio defined through Eq. (2). 73

74

� = ���� (2) 75

76

For δ-1 <8.6E2 whilst for δ-1 >8.6E2, Recrit for a curved tube is equal to that for a straight 77

pipe. 78

79

Another dimensionless number, unique to coiled tubes, is the Dean number, given in 80

Eq. (3). It is used to characterise the flow in curved tubes and quantifies the magnitude of the 81

secondary flow due to the centrifugal force (Mohammed and Narrein, 2012). 82

83

�� = ��√� (3) 84

85

The performance of coiled tubes is a complex function of the coil design parameters 86

(Fig. 1) as well as the resultant pressure drop. Therefore, drag reduction (DR) techniques 87

could be particularly beneficial for systems with curved tubes. Intriguingly, whilst numerous 88

investigations have been reported on DR in straight channels and pipelines with the: injection 89

of air bubbles (Nouri et al., 2013; Fujiwara et al., 2004), dispersion of surfactants (Gasljevic 90

and Matthys, 1997) and polymers (Wei and Willmarth, 1992; Al-Sarkhi and Hanratty, 2001), 91

there is a paucity of research in the field of curved tubes. Moreover, researchers have 92

reviewed the frictional DR techniques in straight channels and pipes (Merkle and Deutsch, 93

1992; Al-Sarkhi, 2010; Murai, 2014) whilst the sole study that reviewed DR in curved tubes 94

was presented by Broniarz-Press et al. (2007). However, the latter focussed on the application 95

of DR surfactant and polymer additives and hence, did not provide a holistic review of the 96

relevant studies. The aim of the current study is to critically review the experimental and 97

numerical studies done on DR in single-phase (water) laminar and turbulent flow through 98

coiled tubes. Such studies are categorized in three sections, representing the pertinent 99

techniques reported. Moreover, this paper complements the earlier review undertaken by the 100

authors of the present study (Fsadni and Whitty, 2016), as it further elucidates the 101

underpinning physics of air-water bubbly flow through curved tubes. It is the authors’ hope 102

that this review will be useful to both academics and industry based engineers through the 103

provision of a concise report on the relevant current knowledge. 104

105

2. Injection of air bubbles 106

107

Over the past 40 years, the injection of microbubbles in the turbulent boundary layer 108

has been investigated by numerous investigators, with the first study reported by McCormick 109

4

and Bhattacharyya (1973) who investigated the DR to a submersible hull. As summarised in 110

Table 1, Shatat et al. (2009a&b) were the first to investigate DR with the injection of air 111

bubbles in laminar and turbulent low through helically coiled tubes. They reported a 112

diminished DR efficiency (Eq. (4)) over that of straight tubes. Such results were more 113

significant with higher curvature ratios whilst, the DR increased with higher air volumetric 114

void fractions (VF) and decreased with higher Re numbers (Fig. 2). Moreover, DR was 115

limited to turbulent flow. Similar results were reported by Saffari et al. (2013) who measured 116

a 25% DR at a VF of 0.09 in turbulent flow bubbly flow. The latter study did not investigate 117

the DR with straight tubes. However, their experimental parameters are comparable to those 118

used by Nouri et al. (2013) who reported a DR of 35% for a VF of 0.09 in straight tubes. 119

�� = 100��������� � (4) 120

121

where fl is the Fanning friction factor for single-phase flow and ftp is the friction factor for 122

two-phase flow. 123

124

For a straight vertical pipe, Fujiwara et al. (2004) reported that, with a high VF in the 125

near-wall region, the turbulence intensity and Reynolds stress are reduced in a wide region of 126

the pipe. The turbulence energy dissipation occurs around the bubbles due to bubble-induced 127

eddies, whilst the diminished fluid density in the near-wall region reduces the shear stress, 128

thus resulting in a lower system frictional pressure drop. Saffari et al. (2013) reported that in 129

curved tubes, higher Re numbers and curvature ratios, result in larger centrifugal forces 130

which force the lower density phase (air bubbles) to migrate towards the inner tube wall 131

region. Resultantly, the shear stress at the inner tube wall region is lower than that at the outer 132

wall region. Hence, the uneven distribution of the air bubbles at higher Re numbers and 133

curvature ratios results in a diminished DR efficiency. 134

135

136

Figure 2: DR as a function of the air VF (α) for a curvature ratio of 0.025 (Shatat et al., 2009a. Fig. 11). 137

These studies are in a general agreement with relevant theory and numerous DR 138

studies reported for channel and straight tube flow. Moreover, there is significant scope for 139

further research in DR (in coiled tubes) as a function of the bubble diameter. In fact, for 140

5

straight tubes and channels, some controversy surrounds the impact of bubble size on the DR, 141

where some investigators reported the DR to be a strong function of the bubble diameter (Liu 142

1993; Murai et al., 2007) while other investigators reported the DR to be independent of the 143

bubble diameter (Moriguchi and Kato, 2002; Shen et al., 2006). The relation of the bubble 144

induced DR studies with those reviewed for two-phase gas-liquid frictional pressure drop 145

characteristics in coiled tubes (Fsadni and Whitty, 2016) remains indeterminate. In fact, the 146

latter investigations reported a general agreement with the Lockhart and Martinelli 147

correlation for straight tubes, with the two-phase frictional pressure drop multiplier in excess 148

of unity. 149

150

3. Surfactant additives 151

152

Surface-active agents (surfactants) are low molecular weight, viscous, non-polymer, 153

water-based chemicals that tend to accumulate at a surface and diminish interactive forces 154

between the molecules of the base fluid, thus reducing the surface tension. Inaba et al. (2005) 155

reported that surfactant additives form a network structure of rod-like micelles which absorbs 156

the turbulent energy with its flexibility and deformation, thus leading to a flow laminarisation 157

effect. Hence, surfactants enhance the elastic properties of the fluid with the resultant 158

increase in DR. Unlike polymer based fluids, the mechanical degradation of the micelle 159

network at high shear stresses is completely reversed at a low flow rate. All the studies 160

reviewed reported a DR limited to the transition and turbulent flows, with a reduced DR in 161

curved tubes when compared to straight tubes, ceteris paribus. Such findings were attributed 162

to the formation of the secondary flow which is largely unaffected by the surfactant additive. 163

Gasljevic and Matthys (1999) reported that for a velocity range of 2-5m/s, the secondary flow 164

effects were separated from the turbulence effects through the use of the turbulence reduction 165

– drag (TRD) method given in Eq. (5). This yielded a TDR of 70% (turbulence suppression) 166

for both coiled and straight tubes (Fig. 3). In contrast, Broniarz-Press et al. (2003) reported 167

that the tube curvature effect on the friction factor was diminished due to the damping of the 168

secondary flows streams. A broad analogy can be made with nanofluid flow in coiled tubes 169

where, nanoparticles were also attributed to the mitigation of the secondary flow (Fsadni and 170

Whitty, 2016). 171

172

��� = � !",�!�� $%&� !",�!�� !",�' (5) 173

174

where lm refers to the laminar flow of the base fluid (without the DR additives) at the same 175

Re number and tb refers to the turbulent flow of the base fluid. 176

6

177

Figure 3: Friction reduction in terms of DR and TRD for a coiled and straight pipe (Gasljevic and 178

Matthys, 1999 Fig. 4). 179

At laminar flow conditions, Weber et al. (1991) and Gasljevic and Matthys (2009) 180

reported an increase in the frictional pressure drop (compared to water). This was attributed 181

to the enhanced solution viscosity. There is a general agreement amongst the studies 182

reviewed that lower coil curvatures and higher surfactant concentrations yielded higher DR 183

efficiencies. Moreover, Kamel and Shah (2013) reported that at higher concentrations, 184

surfactant solutions are more resistant to mechanical degradation and hence, yield higher DR 185

efficiencies at increased Re numbers. Therefore, Broniarz-Press et al. (2002) reported that 186

DR is a strong function of the surfactant concentration, with DR evident above a critical 187

concentration. Inaba et al. (2005) reported that the dynamic nature of surfactant DR additives 188

render them particularly relevant for heating systems. However, such comments should be 189

considered in light of the fact that these additives are known to yield reduced heat transfer 190

coefficients. Kostic (1994) attributed this phenomenon to the non-homogenous turbulence 191

resulting from the flow-induced anisotropicity of the highly structured micelle network. 192

Weber et al. (1991), Inaba et al. (2000&2005), Aly et al. (2006) and Kamel and Shah (2013) 193

presented correlations for the calculation of the friction factor in surfactant solutions. Due to 194

the Non-Newtonian properties of these solutions (C>3,000ppm), correlations were developed 195

as a function of the modified or generalised Re and De numbers. 196

197

4. Polymers additives 198

199

Toms (1948) reported that the addition of minute concentrations of high-molecular 200

weight, long chain and flexible polymers to a Newtonian solvent can yield significant DR 201

properties. Whilst it is widely accepted that the DR efficiency is a strong function molecular 202

weight and distribution, molecular structure and solubility, the underpinning physics are 203

known to be complex and not well-understood (Gallego and Shah, 2009). Factors such as 204

shear thinning, viscoelasticity and molecular stretching have been suggested to diminish the 205

turbulence in the fluid (Bird et al., 1987), thus resulting in DR. 206

Shah and Zhou (2001) stated that the DR mechanism of polymers occurs at the 207

boundary layer and therefore is typically more effective in smaller tube diameters. Moreover, 208

7

in agreement with the findings reported for air-bubble injection, DR efficiency decreases with 209

higher coil curvatures. This is inherent to the effects of the centrifugal force on the fluid 210

flow. DR is also a function of the ability of the polymer to resist thermal and mechanical 211

degradation. Shah et al. (2006) reported that at a volume concentration of 0.07%, the widely 212

used partially hydrolysed polyacrylamide (PHPA) copolymer (Nalco ASP-820) yielded the 213

highest DR (65%). At this concentration, it was assumed that the fluid behaviour is quasi-214

Newtonian. This concentration was subsequently used by Gallego and Shah (2009) and 215

Ahmed Kamel (2011). Gallego and Shah presented a unique generalised friction pressure 216

correlation for DR polymer solutions in coiled tubes. Their correlation assumed that the 217

appropriate characteristic polymer solution viscosity is relative to the zero shear rate 218

viscosity, that is, the shear stress required to deform the polymer molecule from its 219

equilibrium state. 220

The effect of the polymer concentration is also function of the specific physical 221

conditions of the flow. Resultantly, Shah and Zhou (2001) reported that for large tubes and 222

low flow rates, high concentrations of polymer additives increased the fluid drag and delayed 223

the onset of DR (Fig. 4). For small diameter tubes, the opposite effect was reported and thus, 224

a higher polymer concentration increased the DR. 225

226

227 Figure 4: Effect of polymer concentration (Xanthan) on DR ratio (Shah and Zhou, 2001 Fig. 5). 228

The effect of elevated temperatures on the DR of polymers in coiled tubes was 229

investigated by Gallego and Shah (2009) and Ahmed Kamel (2011) who reported that, in 230

contrast to the findings for straight tubes, DR remained quasi-constant (Ahmed Kamel) or 231

increased (Gallego and Shah) with temperature. It is widely accepted that with polymer 232

solutions in straight tubes, elevated temperatures yield a drop in the DR. This is due to a 233

combination of factors, such as the deterioration of the solvent-polymer interaction and the 234

diminishing of the macromolecule size (Clifford and Sorbie, 1985; Nesyn et al., 1989). In 235

view of this complexity and the paucity of studies for curved tubes, Gallego and Shah (2009) 236

and Ahmed Kamel (2011) concluded that the origins of their results are indeterminate and 237

thus require further investigation. In contrast to the numerous studies on polymer DR 238

additives to gas-liquid flows in straight tubes (Sylvester and Brill, 1976; Al-Sarkhi and 239

8

Soleimani, 2004), there are no related studies for coiled tubes. This presents further scope for 240

future research in the field of two-phase flow in coiled tubes. 241

242

Investigators

&

Methodology

Year Flow configuration &

coil geometry

Mean bubble

size

Void fraction

or

concentration

Drag reduction

Air bubbles

Shatat et al.

Experimental

2009a

&b

dt=20mm Dc=800,400,200mm

δ=0.025,0.05,0.1 H=40mm

1,000<Re<100,000 We<1.0

Laminar and turbulent

bubbly flow

db,m=0.06mm

db,max=0.174m

m

No

deformation of

bubbles.

0.21<VF<0.44%

16% for δ=0.025. For a

straight pipe 51% DR,

ceteris paribus. DR effect starts at the

critical Re number.

DR increases with VF

for all cases. The curvature of the coils had a negative

effect on drag reduction.

The Re number corresponding to the

maximum DR was

shifted to a higher value (compared to a straight

tube). This shift increased with an

increase in the curvature of the coil.

Saffari et al.

Experimental

2013 dt=12,19mm

Dc=200mm δ=0.06,0.095

H=24mm

P=0.101MPa 10,000<Re<50,000

Turbulent bubbly flow

db,m=0.27mm

Bubble

diameter

decreased at

higher Re

numbers. At

lower Re

numbers,

bubbles were

less spherical

in shape (less rigid). This is

due to the

influence of

flow stress and

reduced

surface

tension (in

comparison to

the smaller

bubbles).

0.01<VF<0.09

DR increased with

VF with a maximum

of 25% at a VF of

9%. DR diminished

with higher Re

numbers.

At a low VF of 1%, a

DR of 9% was

measured.

DR diminished with

an increase in the curvature of the coil.

Saffari and

Moosavi

Numerical

(Eulerian-

Eulerian

multiphase

model)

2014 dt=16,25,40mm Dc=100,200mm

δ=0.08,0.125,0.20 H=20,60

15,000<Re<80,000

Turbulent bubbly flow

db,m=0.1mm

No

deformation of bubbles.

0.01<VF<0.09

Due to a reduction in

the mixture density,

higher VF yields lower pressure drops,

shear stress and

friction coefficient.

9

Surfactant solutions & Foam fluids

Weber et al.

Experimental

1991 dt=10.5,16.5mm 157<Dc<454mm

0.105<δ<0.036, N=12,18,34,39

1,500<Re<100,000

6,750<Recrit<9,480

30oC<T<90oC

Laminar and turbulent

Fluid was

assumed to be

quasi-

Newtonian.

C=62.5;250;1,

000 ppm

Habon in

water.

For laminar flow,

surfactant additives

increased the fluid

drag. For turbulent flow the

increase in DR with

C was marginal.

DR in curved tubes

diminished at a lower

Re value than that in

straight tube, ceteris

paribus.

(�)**�*+ = 1855����� + 0.011

Gasljevic and

Matthys

Experimental

1999 dt=2mm Dc=200mm

δ=0.01 1.8<V<7m/s

T=25oC

Laminar and turbulent

Fluid was

assumed to be

quasi-

Newtonian.

C=2,000ppm

SPE95285

(Same

viscosity as

water)

DR in coiled tube is

30%, in a straight

tube 60%, ceteris

paribus.

Calculated 70%

reduction in turbulence effects for

both straight and

coiled tubes.

At V>5m/s DR effect

diminishes due to

micelle degradation.

Inaba et al.

Experimental

2000 dt=17.7mm

Dc=177,300.9,442.5,885mm

δ=0.02,0.04,0.059,0.1 400<Re’<200,000

10oC<T<25oC

θ=45o,90o,180o,270o

Laminar and turbulent

Non-

Newtonian

viscoelastic

fluid.

530<C<1,773

ppm

Dodecyltrimet

hyl

Ammonium Chloride

(C12H25N(CH3

)3 =263.89)

and Sodium

Salicylate

(C7H5NaO3=1

60.10) in

water

No DR at laminar

flow conditions,

whilst DR at

turbulent flow

conditions was less in relation to that in a

straight pipe.

At a C of 561ppm no

DR was measured.

Due to the

suppression of

turbulence vortexes,

the heat transfer

coefficient was less

than that for water.

(�)��/ = 6.75 2��3� 4��.56� 7�.896��′��.5

0.02<δ<0.05; 45o<θ<270o; C>1,000ppm

(SD=9.17%)

Broniarz-Press

et al.

Experimental

2002 0.0219<δ<0.0792 1,200<Regen<30,000

70<De’’<3,000 T=303,323,333K

Laminar and turbulent

Non-

Newtonian

viscoelastic

fluid.

WC=0.1,0.25

%

Cationic

Hexadecyltrim

ethylammoniu

m chloride

(HTAC) and

DR is only evident

above a critical C.

This contrasts to

polymers where DR

is significant with

minute C of polymer

additives.

10

anionic soaps,

sodium &

potassium

oleates with

WC=2.5,7%

sodium

salicylate

(NaSal),

sodium

chloride, and

potassium

chloride

solution

additives in

water.

With polymer

additives, DR is only

evident when the

molar mass is above a

critical value.

Cylindrical micelles

stabilise the

mechanisms of

curved flow.

DR increases with

higher turbulence.

Broniarz-Press

et al.

Experimental

2003 0.0219<δ<0.0792 1,200<Regen<30,000

70<De’’<3,000 T=303,313,333K

Laminar and turbulent

Non-

Newtonian

viscoelastic

fluid.

WC=0.1,0.25

%

Cationic

Hexadecyltrim

ethylammonium chloride

(HTAC) and

anionic soaps,

sodium &

potassium

oleates with

WC=2.5,7%

sodium

salicylate

(NaSal),

sodium chloride, and

potassium

chloride

solution

additives in

water.

DR observed in

turbulent and

pseudolaminar flows.

Surfactant additives

diminished the tube curvature effect on

the friction factor.

This was attributed to

the damping of the

secondary flow

streams.

Inaba et al.

Experimental

2005 dt=14.4mm

Dc=540mm δ=0.0267

H=32mm

N=10 10,000<Re’<100,000

100<De/De’<10,000 100<Gz/Gz’<10,000

5oC<T<20oC Laminar and turbulent

Non-

Newtonian

viscoelastic

behaviour at

high

concentrations (>3,000ppm)

1,000<C<3,50

00ppm

Mixture of

oleyldihydrox

yethylamineox

ide (ODEAO, C22H45NO3=3

71) 90%, non-

ionic

surfactant &

cetyldimethyla

minoaciticacid

betaine

(CDMB,

C20H41NO2=3

27) 10% as a

zwitterion

surfactant in water.

43% DR in the coiled

tube.

77% DR in a straight

tube.

This is due to the

secondary flow that contributes towards

the pressure drop in

coiled tubes.

Drop in the heat

transfer coefficient

with surfactant C.

DR increases with

surfactant C.

11

(�,�)��/(;�,�)��/ = ��<�.9�=*��.88�*�8.5

where:

�*� = >?��@?�>�AB�B�?�(DEFG); =*� = ?��@?�

�AB�B�?�(J,KKK��')

Aly et al.

Experimental

2006 dt=14.4mm Dc=320,540,800,mm

0.018<δ<0.045 H=32mm

N=10

1,000<Re’<100,000 5oC<T<20oC

Laminar and turbulent

Newtonian

fluids for

C<3,000ppm.

250<C<5,000

ppm

Mixture of

non-ionic

surfactant oleyldihydrox

yethylamineox

ide (ODEAO,

C22H45NO3=3

71) 90%, &

cetyldimethyla

minoaciticacid

betaine

(CDMB,

C20H41NO2=3

27) 10% as a

zwitterion surfactant in

water.

DR increased with

surfactant C, with a

max. of 59% at

Re’=55,350 and

C=2678ppm. DR increased with

temperature and

decreased with higher

coil curvatures.

Lower DR and losses

in the heat transfer

coefficient were

measured when

compared to straight

tubes, ceteris paribus.

(�)��/ = 137��.6�(1 + 0.94=N��.�9�N�8.5O)(1.56 + PQR��′)5.O�

(SD=10%)

1<Tc<1.065; 4<Cc<14; 0.018<δ<0.045

Gasljevic and

Matthys

Experimental

2009 dt=12mm

δ=0.043,0.067,0.116 0.9<V<7m/s

T=25oC

Laminar and turbulent

Non-

Newtonian

viscoelastic

fluid.

C=2,000ppm

cationic

surfactant

Ethoquad T-

13 &

2,000ppm

NaSl as a counterion.

DR for turbulent flow

in the range of 30-

40% was measured.

This is less that that

in a straight pipe

where 75% DR was

measured, ceteris paribus.

DR decreased with

higher curvature

ratios.

For the coil with the

highest curvature, at

V=0.9m/s, the

pressure drop

increased in relation

to that of water. This

was attributed to the

higher viscosity of the surfactant

solution in relation to

water at a shear rate

of 500s-1

.

Kamel and

Shah

Experimental

2013 11.0<dt<63.5mm 360<Dc<2,850mm

0.01<δ<0.031 20,000<Re<200,000

Turbulent

Non-

Newtonian

viscoelastic

fluid.

VC=1.5,2.3,4

%

Tallowalkyla

midopropyl

DR is significant in

coiled tubes and

increases with C,

with a significant

12

dimethylamine

oxide

viscoelastic

surfactant

(VES)

containing 50-

65% WC

active

surfactant, 25-

40%

propylene

glycol and

water as

solvents.

increase above a VC

of 2%. Higher C also

exhibit higher

resistance to

mechanical

degradation.

Surfactant based

fluids are more

resistant to shear

degradation than

polymer based fluids.

Larger tube diameters

and smaller curvature

ratios yield larger

DR.

(�)**�*+ = (−32,200.42�� + 1,830.62�� + 0.32)��+T*UO,�8�.V5WX��86.VOW��.55Y

where:

��+T* = Z3�*[��*\8*�8] ^

Wang et al.

Numerical

2015 dt=7.3mm

Dc=203mm δ=0.036

V=3m/s

Compressible

Non-

Newtonian

foam fluid.

65<Γ<98 The secondary flow

effect (vortex roll) of

the foam fluid is

smaller than that of

water.

Polymer solutions

Barnes and

Walters

Experimental

1969

dt=8,9.6mm 60<Dc<3000mm

0<Q<80cm3/s T=20oC

Spiral coil Laminar and turbulent

Non-

Newtonian

viscoelastic

fluid.

Solvent: Water

VC=0.025,0.0

3,0.05,0.10%

Polyacrylamid

e (P250);

Polyethylene oxide (Polyox

SR305) and

Guar Gum.

Easier to pump

viscoelastic liquids in

curved tubes.

Suppression of

turbulence with polymer additives

which renders the

flow almost laminar.

Curvature enhances

DR in the transition

region, whilst it

reduces DR at high

Re numbers.

Kelkar and

Mashelkar

Experimental

1972 dt=12.5mm

Dc=665mm δ=0.019

H=38mm N=6

10<Re<100,000

Laminar and turbulent

Non-

Newtonian

viscoelastic

fluid. Solvent:

Water

50<C<500pp

m

Polyacrylamid

e (AP30&ET59

7)

0.76<n<1.00

DR limited to

turbulent flow. DR

increases with

polymer C up to a critical Re when DR

diminishes.

_ = 0.2 + 0.81 + (`�T< )�.a

where:

�̀T< = � b$��cTK.EF

Z� b$��cTK.EF^dAeK.f

; _ = �.6 ;�

Mashelkar and

Devarajan

Experimental

1976 dt=12.48,12.49,12.50mm 92.3<Dc<1,282mm

0.01<δ<0.135 H=38.1mm

3<N<40

Non-

Newtonian

viscoelastic

fluid. Solvent:

0.01<C<0.5%

Carboxymethy

l cellulose

(CMC), Polyacrylamid

The PEO and PAA

polymer yielded the

best DR, even at the

lowest C. This was attributed to the fluid

13

10<Regen<100,000 70<De<400

40<Wi<950 Laminar and turbulent

Water e (PAA-AP-

30)

0.354<n<0.99

Polyethylene

oxide (PEO-

WSR-301)

0.871<n<0.99

elasticity.

(g,�)**�*+ = (;(1 − 0.03923hi�.�9aa) where: (;,�)**�*+ = (9.069 − 9.438j + 4.374j�)��.5��′′(��.O6ak�.8��*)

0.35<n<1

Oliver and

Asghar

Experimental

1976 6.72<dt<14.0mm

0.033<δ<0.082 152<L/dt<410

N=3-4 60<De<2,000

10<Gz<400 Laminar

Non-

Newtonian

viscoelastic

fluid.

Solvent:

Water.

250<C<2,500

ppm

Polyacrylamid

e Separan

AP273 in

water and a

56/44 (WC)

glycerol/water

solution with

500ppm

Separan

AP273.

Some DR due to the

partial suppression of

the secondary flow.

Rao

Experimental

1993

dt=9.35mm 98<Dc<247mm

0.038<δ<0.095 H=19.5mm

8<N<20

10,000<Re<60,000 Turbulent

Non-Newtonian

viscoelastic

fluid.

Solvent:

Water

C=50,100,200ppm

Polyacrylamid

e (Praestol

2273TR)

Higher DR with higher polymer C and

smaller coil

curvatures.

Azouz et al.

Experimental

1998 dt=30mm pH=9,10,11

100<Regen<100,000 Laminar and Turbulent

Non-Newtonian

viscoelastic

fluid.

Solvent:

Water

C=35,40 lb/kgal

Linear Guar

gum &

Hydroxypropy

l Guar (HPG),

Crosslinked

Guar gum &

Hydroxypropy

l Guar (HPG)

with 12% sol.

of boric acid

as crosslinking agent.

For borate-crosslinked HPG, the

pressure gradient is a

strong function of pH

and the tube length.

For borate

crosslinked guar, the

pressure gradient is

pH dependent but is

not effected by the

tube length.

14

Shah and

Zhou

Experimental

2001 dt=25.4,38.1,60.3mm Dc=121.92,182.88,281.94

mm δ=0.0113,0.0165,0.0169

Pmax=34.47MPa 4,000<Regen<200,000

Laminar and Turbulent

Non-

Newtonian

viscoelastic

fluid.

Solvent:

Water

Guar

C=2.397

kg/m3

0.642<n<0.72

C=3.595

kg/m3

0.527<n<0.55

C=4.793

kg/m3

0.433<n<0.48

3

partially

hydrolysed

polyacrylamid

e (PHPA),

C= 2.397

kg/m3

0.355<n<0.38

4 C=4.793

kg/m3

0.305<n<0.32

2

Xathan gum

C=1.198

0.472<n<0.48

9

C=2.397

0.381<n<0.43

9 C=4.793

0.277<n<0.34

3

hydroxyethylc

ellulose

(HEC)

C= 2.397

0.6<n<0.668

C=3.595

0.494<n<0.54

5 C=4.793

0.42<n<0.443

DR of polymer

solutions decreases

with the curvature

ratio.

Xathan and PHPA

yielded the best DR

properties. HEC

resulted in no DR.

Higher DR with

smallest tube

diameters.

For the largest tube

diameter, higher

polymer C decreased

the onset of the DR

whilst the opposite

effect was reported

for the smallest tube

diameter.

Shah et al.

Experimental

2006 dt=11mm

Dc=35.60,57.24,109.97mm

δ=0.01,0.019,0.031

N=3,6 22,000<Res<155,000

Turbulent

For

0.01<C<0.07

% fluid is

assumed to be

Newtonian.

Non-

Newtonian

viscoelastic

fluid for

C>0.07%.

Solvent: Water

Nalco ASP-

820 (PHPA)

0.01<VC<0.1

5%

0.814<n<1.00

Optimum VC of

ASP-820 is 0.07%.

At 0.07%, ASP-820

yields a DR of 75%

in straight tube and

65% in coiled tube,

ceteris paribus.

Increase in flow rate

increases the DR

while the opposite

effect was reported for an increase in

curvature.

An increase in the

polymer C or

curvature ratio delays

the onset if DR.

15

(g,�)**�*+ = l<�mn 2 8.�cTopn4 where A’,B’&C’ are constants given in Shah and Ahmed

Kamel, (2005) and is valid for VC=0.07%.

(ME= ±6%)

Zhou et al.

Experimental

2006 dt=11.05mm Dc=12.14,29.67,47.70,91.

64mm δ=0.010,0.019,0.031,0.07

6

N=3,6,7 5,000<Regen<100,000

Laminar and turbulent

Non-

Newtonian

viscoelastic fluid.

Solvent:

Water

C=10,20,30

lb/Mgal

Guar gum, C=10,15,20,30

lb/Mgal

Hydroxypropy

l Guar (HPG),

C=10,20,30

lb/Mgal

Xanthan gum

DR in coiled tubing is

diminished (by 10-

30%) in relation to that in a straight tube,

ceteris paribus.

DR in coiled tubing is

increased with higher

Re. This contrasts to

the case of straight

tubes, where DR

diminishes at higher

Re.

DR increased with C

of Xanthan. Curvature delayed the

onset of DR as a

result of the delay in

the onset of

turbulence.

Gallego and

Shah

Experimental

2009 dt=11,20.57mm

Dc=35.60,57.24,109.97,182.88cm

δ=0.01, 0.0113,

0.019,0.031 22,000<Res<430,000

T=21.1,37.7,54.4oC Turbulent

For

0.01<C<0.07

% fluid is

assumed to be

Newtonian.

Non-

Newtonian viscoelastic

fluid for

C>0.07%.

Solvent:

Water

Nalco ASP-

700 & ASP-

820 (PHPA)

VC=0.05,0.07,

0.10,0.15%

0.75<n<1.00

DR decreases with

curvature.

DR in coiled tubes is

lower than that in

straight tubes, ceteris

paribus. At 0.07%

ASP-820, DR is 77% in a straight tube and

64% in the coiled

tube (79%&59% for

ASP-700).

The increase in T

resulted in a decrease

of DR in straight

tubes. The opposite

effect was measured

in coiled tubes

(DR=45%,52%&55% at 21.1,37.7,54.4

oC

respectively for ASP-

820)

DR decreases with

tube roughness in

both straight and

coiled tubes (64% to

60% for coiled tube).

`�T =qrrrs 1.6675 ∗ 10��u(;,�)**�*+��;v8.9�a9 �8wx3� �Z1 + 1.0974 ∗ 10�� �(;,�)**�*+��; 8wx3� �

8.9���5^�.O588

yzzz{|\g};\g}~�

�.88�V

16

`�T = Z(;,�)**�*+(g,�)**�*+^�− 1

(ME= ±10%)

Shah and

Zhou

Experimental

2009 dt=12mm Dc=146,356,572, 1100

mm δ=0.01,0.019,0.031,0.076

N=3,3,7

3,700<Regen<11,500 Laminar and turbulent

Non-

Newtonian

viscoelastic

fluid.

Solvent:

Water

1.198<C<3.59

5 kg/m3

Guar gum,

0.482<n<0.81

9

Hydroxypropy

l Guar (HPG),

0.485<n<0.80

5

Xanthan gum 0.310<n<0.71

7

Significant DR with

all three polymer

fluids. Curvature

reduces the DR and

delays the onset of

DR.

1�(�)**�*+ =

10.05311 + 0.29465��.5 PQR8� 2��+T*(

8�4 + 10.03094 + 0.24575��.5

Ogugbue and

Shah

Numerical

2011 δ=0.3,0.5,0.6,0.8

ε=0,0.25,0.5,0.75,0.96 100<Regen<10,000

Laminar and turbulent

Non-

Newtonian

viscoelastic

fluid.

Solvent:

Water

C=20,30,40,60

lb/Mgal

Guar

0.335<n<0.66

6

DR increases with

increased eccentricity

(50% reduction for

fully eccentric

annular section)

Higher C increased

the frictional pressure

drop for laminar flow.

For turbulent flow, all

C resulted in a

significant DR.

(�)**�*+ = 0.00378 3T��3�~� +3.7374��+T* + 4042

2��+T* −0.00124

(ME= ±5%)

Ahmed Kamel

Experimental

2011 dt=11mm Dc=579mm δ=0.019

T=22,35,38oC 20,000<Re<200,000

Pmax=6.9MPa Turbulent

Properties

assumed to be

quasi-

Newtonian. Solvent:

Water

Nalco ASP-

820 (PHPA)

VC=0.07%

n ≈ 1.00

DR in the range of

30-80%

At elevated T, the DR

effect is diminished in straight tubes

while it remains

quasi- constant in

coiled tubes.

��>��) = 1.0

(ME= ±2.1%)

Table 1: Review of the experimental and numerical work 243

244

5. Conclusions 245

246

17

The studies reviewed have demonstrated that, due to the secondary flow, which 247

increases with curvature, DR in coiled tubes is diminished when compared to straight tubes. 248

However, a significant DR can be still be achieved with the introduction of: bubbles (9-25%), 249

surfactant (30-59%) and polymer (circa 30-80%) additives. DR is a strong function of the 250

surfactant concentration and the air volume fraction whilst with polymer additives DR 251

efficiency is dependent on the molecular weight, structure and solubility. DR is generally 252

present in flows with Re numbers in excess of the critical number. However, at elevated Re 253

numbers DR diminishes. This is due to the higher centrifugal forces (air bubbles and 254

polymers) and mechanical degradation with high shear stress (surfactants). A number of 255

authors have presented correlations for the calculation of the friction factor which are 256

typically a function of the: curvature ratio, Re and De numbers and the additive 257

concentration. 258

Due to their low molecular weights, viscous properties and resilience to mechanical 259

degradation, surfactant based fluids are generally considered to be superior to polymer based 260

fluids. Hence, surfactants are suitable for a variety of applications such as district cooling and 261

heating systems. A significant scope for future research has been elucidated for DR in coiled 262

tubes with the injection of air bubbles (impact of bubble size and relation with the Lockhart 263

and Martinelli correlation) and the application of a combination of methods, such as the use 264

of polymer and surfactant additives with bubbly flow. 265

266

Acknowledgments 267

268

The authors of the current investigation would like to thank the University of Central 269

Lancashire UK, for facilitating the completion of this study as well as the various authors 270

who have been contacted during the course of this study. 271

272

Notation List 273

274

C concentration (ppm) 275

Cc non-dimensional surfactant concentration (-) 276

Cst empirical constant (-) 277

d tube internal diameter (m) 278

dr drag ratio (-) 279

D helix diameter (m) 280

De Dean number (Reδ1/2

) (-) 281

De’ modified Dean number (Re’δ 1/2

) (-) 282

De’’ modified Dean number (Regenδ1/2

) (-) 283

DR drag reduction (%) 284

f friction factor (-) 285

FC friction coefficient (-) 286

Gz Graetz number (RePr/z) (-) 287

Gz’ modified Graetz number (Re’Pr’/z) (-) 288

H pitch (m) 289

K rheometric and technical consistency index (Pa sn) 290

L length (m) 291

ME mean error (%) 292

n power law model flow behaviour index (-) 293

N number of turns (-) 294

NDe Deborah number (-) 295

NDe’ modified Deborah number (-) 296

18

P pressure (Pa) 297

Pr Prandtl number (-) 298

Pr’ modified Prandtl number (-) 299

Q volume flow rate (m3/s) 300

Re Reynolds number (-) 301

Re’ modified Reynolds number as proposed by Metzner and Reed (1955) 302

�88�* ��*k89* � ��D������� �� (-) 303

Recrit critical Reynolds number (24��.��) (-) 304

Regen generalised Reynolds number ��D������a��J� � (-) 305

SD standard deviation (%) 306

T temperature (0C) 307

Tc non-dimensional surfactant solution temperature (-) 308

TRD turbulence reduction: drag (-) 309

V flow velocity (m/s) 310

VC volume concentration (%) 311

VF volumetric void fraction (-) 312

We Weber number (-) 313

Wi Weissenberg number (σel/σv) (-) 314

WC weight concentration (%) 315

x axial distance of coiled pipe (m) 316

z dimensionless axial distance (x/dt) (-) 317

318

Greek 319

320

β reduced friction factor (-) 321

δ curvature ratio (-) 322

ε coil eccentricity (-) 323

θ angle from inlet of curved pipe (o) 324

λ relaxation time (s) 325

µ viscosity (cP) 326

µo zero shear rate viscosity (cP) 327

ν average fluid velocity (ft/s) 328

ρ density (kg/m3) 329

σ stress (N/m2) 330

Γ quality (%) 331

332

Subscripts 333

334

a ambient temp 335

b bubble 336

bf base fluid 337

c coil 338

crit critical 339

DRF drag reducing fluid 340

eff effective 341

el elastic 342

eit external diameter of inner tubing 343

gen generalised 344

iot internal diameter of outer tubing 345

19

l liquid 346

lm laminar 347

m mean 348

nd non-dimensional 349

o zero 350

p polymer solution 351

s solvent 352

st straight tube 353

t tube 354

tb turbulent 355

tp two-phase 356

T elevated temperature 357

v viscous 358

359

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Highlights

Review on pressure drop reduction studies in helically coiled tubes

Air bubbles, surfactant and polymer additives are effective in diminishing drag

Drag reduction is diminished in relation to straight tubes

Drag reduction is predominantly evident in turbulent flow

Drag reduction diminishes with higher coil curvatures and excessive Re numbers

*Highlights (for review)